In Romans chapter 2, the Apostle Paul speaks on the fact that God has wrath on those who do evil, takes pleasure in those who do righteously, and will reward them each accordingly, irrespective of whether they are Jews or Gentiles.
The passage terminates with "for not the hearers of the law are just before God, but the doers of the law shall be justified":
Romans 2:5-13
5 But after thy hardness and impenitent heart treasurest up unto thyself wrath against the day of wrath and revelation of the righteous judgment of God;
6 Who will render to every man according to his deeds:
7 To them who by patient continuance in well doing seek for glory and honour and immortality, eternal life:
8 But unto them that are contentious, and do not obey the truth, but obey unrighteousness, indignation and wrath,
9 Tribulation and anguish, upon every soul of man that doeth evil, of the Jew first, and also of the Gentile;
10 But glory, honour, and peace, to every man that worketh good, to the Jew first, and also to the Gentile:
11 For there is no respect of persons with God.
12 For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law;
13 (For not the hearers of the law are just before God, but the doers of the law shall be justified.
This passage, especially verses 7 and 13, are very commonly used to teach works Salvation, and so we should have a good understanding of what is being said here.
The passage in question forms the second part of the Apostle Paul's opening to the book of Romans, and the thought that he is developing here begins in chapter 1.
In that chapter, he opens with the fact that he doesn't care that the Romans are Gentiles, because of their mutual faith in the same Gospel (Romans 1:12-17).
He then goes on to describe "all ungodliness and unrighteousness of men" (Romans 1:18), which begins with rejecting natural revelation (Romans 1:19-21), and results in every kind of wickedness (Romans 1:22-32).
Then, chapter 2 begins with a condemnation of hypocrisy (Romans 2:1-4), before we arrive at our passage in question, which again, is:
Romans 2:5-13
5 But after thy hardness and impenitent heart treasurest up unto thyself wrath against the day of wrath and revelation of the righteous judgment of God;
6 Who will render to every man according to his deeds:
7 To them who by patient continuance in well doing seek for glory and honour and immortality, eternal life:
8 But unto them that are contentious, and do not obey the truth, but obey unrighteousness, indignation and wrath,
9 Tribulation and anguish, upon every soul of man that doeth evil, of the Jew first, and also of the Gentile;
10 But glory, honour, and peace, to every man that worketh good, to the Jew first, and also to the Gentile:
11 For there is no respect of persons with God.
12 For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law;
13 (For not the hearers of the law are just before God, but the doers of the law shall be justified.
Above, the Apostle Paul states, in essence, that God is a righteous judge, who "will render to every man according to his deeds" (Romans 2:6). As such, He will genuinely grant "Eternal Life" to those who "patiently continue in good works" (Romans 2:7), and that "the doers of the Law shall be justified" (Romans 2:13) before God on that basis.
And this is where the works Salvationist stops reading, as that is the only possible way to take this passage as a genuine roadmap for how we are to get "Eternal Life", and be "justified".
This is because the Apostle Paul continues his thought in the next chapter, where he makes it clear that no one - Jew or Gentile - "patiently continues in well doing", or "does the Law":
Romans 3:5-12
5 But if our unrighteousness commend the righteousness of God, what shall we say? Is God unrighteous who taketh vengeance? (I speak as a man)
6 God forbid: for then how shall God judge the world?
7 For if the truth of God hath more abounded through my lie unto his glory; why yet am I also judged as a sinner?
8 And not rather, (as we be slanderously reported, and as some affirm that we say,) Let us do evil, that good may come? whose damnation is just.
9 What then? are we better than they? No, in no wise: for we have before proved both Jews and Gentiles, that they are all under sin;
10 As it is written, There is none righteous, no, not one:
11 There is none that understandeth, there is none that seeketh after God.
12 They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.
Above, we see that "both Jews and Gentiles...are all under sin" (Romans 3:9), and so "there is none righteous, no, not one" (Romans 3:10). Likewise, none "patiently continue in well doing" (Romans 2:7), because "there is none that doeth good, no, not one" (Romans 3:12).
Therefore, the Apostle Paul continues on to explain, the Law was only to reveal our need for the Savior, and it is only by faith in Jesus Christ that we can be justified:
Romans 3:19-28
19 Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God.
20 Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin.
21 But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets;
22 Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference:
23 For all have sinned, and come short of the glory of God;
24 Being justified freely by his grace through the redemption that is in Christ Jesus:
25 Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;
26 To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus.
27 Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith.
28 Therefore we conclude that a man is justified by faith without the deeds of the law.
Above is the powerful "conclusion" (Romans 3:28) to the thought that he has been working on this entire time up to this point in the book. He makes it abundantly clear that no one is justified by keeping the Law, and instead we are "justified by faith, without the deeds of the Law" (Romans 3:28).
The doctrine of justification without the Law is taught repeatedly in Scripture (Acts 13:39, Romans 9:31-32, 10:4, Galatians 2:16, 3:11, 3:21-24, Philippians 3:9, etc.), and the article You Do Not Have to Keep the Law to be Saved also covers the topic in depth. Twisting Romans 2:5-13 to teach that someone has to keep the Law to be saved is simply dishonest.
So, to recap, the outline of Paul's opening to Romans is:
The most common refutation of the argument given above, is to assert that whenever Paul says that we are "justified without the Law", he is referring only to the abrogated aspects of the Mosaic Law, which have been done away with in Christ - circumcision, dietary restrictions, temple sacrifices, clothing requirements, and the like. However, he didn't have in mind the moral part of the Law, and so we still must keep the moral Law to be saved.
This is not true. When Paul says "Law", he means the entire Law, which as demonstrated in the aforementioned article on the Law, encompasses any good thing you can do, and any avoidance of bad that you may do, because it is the very standard of righteousness, as we read in the same book:
Romans 7:7-8
7 What shall we say then? Is the law sin? God forbid. Nay, I had not known sin, but by the law: for I had not known lust, except the law had said, Thou shalt not covet.
8 But sin, taking occasion by the commandment, wrought in me all manner of concupiscence. For without the law sin was dead.
Above, we see the "Law" being used to reference a moral commandment - thou shalt not covet.
Also, note that the passage in question said "the doers of the Law shall be justified" (Romans 2:13). Are we supposed to believe that that was to the exclusion of the moral Law? They are trying to have their cake and eat it too, where the Law contains the moral code when they need the verse to teach works Salvation, but not when the verse says we are justified without the Law.
We can also look at the way that the "Law" is spoken of in the buildup to Paul's conclusion in chapter 3, just after our passage in question:
Romans 2:14-15
14 For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves:
15 Which shew the work of the law written in their hearts, their conscience also bearing witness, and their thoughts the mean while accusing or else excusing one another;)
Here, we see the "Law" being used to refer to what definitely includes moral commandments, as, obviously, Gentiles don't get a strange feeling when wearing mixed fabrics, or being uncircumcised, but every Gentile nation on earth had laws against stealing, killing, and adultery - testifying that God wrote the moral parts of the Law on their heart in the form of a conscience.
There are other statements, just within Romans, that make it evident that when Paul said "Law", he meant "Law", not "certain abrogated aspects of the Law", for example:
Romans 13:8-10
8 Owe no man any thing, but to love one another: for he that loveth another hath fulfilled the law.
9 For this, Thou shalt not commit adultery, Thou shalt not kill, Thou shalt not steal, Thou shalt not bear false witness, Thou shalt not covet; and if there be any other commandment, it is briefly comprehended in this saying, namely, Thou shalt love thy neighbour as thyself.
10 Love worketh no ill to his neighbour: therefore love is the fulfilling of the law.
Above, the Apostle Paul uses the "Law" to refer to explicit moral laws that were part of the Ten Commandments, along with "any other commandment".
Therefore, this argument is just a dishonest attempt to nullify what the Word of God says, in order to teach works Salvation.
In conclusion, this passage in Romans chapter 2 definitely does not teach works Salvation, but rather constitutes part of an argument by the Apostle Paul actively teaching against that doctrine.
It speaks to the fact that God is perfectly just in His judgments, and so in order to rescue us, despite our universal guilt, He give us Jesus Christ as a propitiation for our sins.